# Angular acceleration

Unit systemSI derived unit
Unit ofAngular acceleration

Angular acceleration is the time rate of change of angular velocity. In three dimensions, it is a pseudovector. In SI units, it is measured in radians per second squared (rad/s2), and is usually denoted by the Greek letter alpha (α). Just like angular velocity, there are two types of angular acceleration: spin angular acceleration and orbital angular acceleration, representing the time rate of change of spin angular velocity and orbital angular velocity respectively. Unlike linear acceleration, angular acceleration need not be caused by a net external torque. For example, a figure skater can speed up her rotation (thereby obtaining an angular acceleration) simply by contracting her arms inwards, which involves no external torque.

## Mathematical definition

The angular acceleration vector is defined as:

${\boldsymbol {\alpha }}={\frac {d{\boldsymbol {\omega }}}{dt}}$ ,

where ${\boldsymbol {\omega }}$ can be either the orbital or the spin angular velocity vector, depending on whether ${\boldsymbol {\alpha }}$ is the orbital or spin angular acceleration vector.

### Equation of Motion for a Point Particle

The orbital angular acceleration of a point particle α can be connected to the applied torque τ by the following equation:

${I{\boldsymbol {\alpha }}}={\boldsymbol {\tau }}-{\frac {d{I}}{dt}}{\boldsymbol {\omega }}$ ,

where I is its moment of inertia.

The above relationship indicates that, unlike the relationship between force and acceleration, the orbital angular acceleration need not be directly proportional or even parallel to the torque. However, in the special case where the distance to the origin does not change with time, the torque does turn out to be proportional and parallel to the angular acceleration (with the constant of proportionality being the moment of inertia of the particle).